A Topological characterisation of Weisfeiler-Leman equivalence classes

TL;DR

This paper uses topological covering space theory to characterize the limitations of message passing GNNs in distinguishing non-isomorphic graphs, introduces a large dataset of indistinguishable graphs, and empirically confirms GNNs' inability to differentiate them.

Contribution

It provides a topological framework to fully characterize GNN indistinguishability classes and introduces the GraphCovers dataset of non-distinguishable graphs.

Findings

GNNs cannot distinguish graphs in the GraphCovers dataset

The number of indistinguishable graphs grows super-exponentially with nodes

None of the tested GNN architectures could differentiate graphs in the dataset

Abstract

Graph Neural Networks (GNNs) are learning models aimed at processing graphs and signals on graphs. The most popular and successful GNNs are based on message passing schemes. Such schemes inherently have limited expressive power when it comes to distinguishing two non-isomorphic graphs. In this article, we rely on the theory of covering spaces to fully characterize the classes of graphs that GNNs cannot distinguish. We then generate arbitrarily many non-isomorphic graphs that cannot be distinguished by GNNs, leading to the GraphCovers dataset. We also show that the number of indistinguishable graphs in our dataset grows super-exponentially with the number of nodes. Finally, we test the GraphCovers dataset on several GNN architectures, showing that none of them can distinguish any two graphs it contains.